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Detailed Balance in Ehrenfest Mixed Quantum-Classical Dynamics J. Chem. Theory Comput. 2006, 2, 229-235 229 Detailed Balance in Ehrenfest Mixed Quantum-Classical Dynamics Priya V. Parandekar and John C. Tully* Department of Chemistry, Yale UniVersity, P.O. Box 208107, New HaVen, Connecticut 06520 Received August 25, 2005 Abstract: We examine the equilibrium limits of self-consistent field (Ehrenfest) mixed quantum- classical dynamics. We derive an analytical expression for the equilibrium mean energy of a multistate quantum oscillator coupled to a classical bath. We show that, at long times, for an ergodic system, the mean energy of the quantum subsystem always exceeds the temperature of the classical bath that drives it. Furthermore, the energy becomes larger as the number of states increases and diverges as the number of quantum levels approaches infinity. We verify these results by simulations. 1. Introduction total energy is conserved, and both methods have proved Mixed quantum-classical dynamics (MQCD), in which quite accurate in many applications. selected quantum mechanical degrees of freedom are coupled We showed in a previous paper that, for a two-level to a system of classical mechanical degrees of freedom, has quantum system coupled to a many-particle classical bath, proved to be a useful complement to standard classical the “fewest-switches” version of surface hopping13 correctly molecular dynamics (MD) simulations. Quantum effects obeys detailed balancing; the two-level system approaches including electronic transitions,1,2 proton tunneling, and zero- a quantum temperature equal to the classical temperature of point motion3-6 can be introduced within a computationally the bath.21 This is not necessarily the case for the Ehrenfest tractable classical MD framework. A critical requirement for method, as has been discussed by several authors,21-24 the success of a MQCD theory is the proper treatment of signaling a potentially serious deficiency of the method. We the “quantum backreaction”, the altering of the classical previously derived a closed-form expression for the mean forces due to transitions in the quantum subsystem.7-12 Two energy of a two-level quantum system coupled by Ehrenfest widely used approaches for approximating the quantum dynamics to a classical bath, showing that the quantum backreaction have emerged, “surface hopping”13-15 and subsystem approaches a temperature that is finite but higher “Ehrenfest”.16-20 In both approaches, quantum transitions than the temperature of the classical bath to which it is arise in the same way, governed by the time-dependent coupled.21 In this paper, we generalize our previous result Schro¨dinger equation in which the time variation of the to a quantum subsystem composed of an arbitrary number Hamiltonian arises from the motions of the classical particles. of quantum states. For the special case of equally spaced The methods differ only in the way the classical paths evolve. quantum levels, we are able to obtain an exact, closed-form In surface hopping, the forces derive from a single quantum expression for the mean energy of the quantum subsystem. state, subject to sudden stochastic “hops” to different This expression shows, remarkably, that in the limit of an quantum states. The Ehrenfest method is a self-consistent infinite number of equally spaced levels, that is, a harmonic field method; the forces governing the classical particles arise oscillator, the mean energy of the quantum subsystem from a weighted average of quantum states. Both the surface- approaches infinity no matter how low the classical temper- hopping and Ehrenfest methods allow for energy transfer ature that drives it. The populations of each state are not between the quantum and classical subsystems such that the equal in this limit, so the quantum subsystem does not approach infinite temperature. However, the populations * Corresponding author phone: (203) 432-3934; fax: (203) 432- decrease with increasing quantum number sufficiently slowly 6144; e-mail: [email protected]. so that the mean energy diverges. 10.1021/ct050213k CCC: $33.50 © 2006 American Chemical Society Published on Web 01/10/2006 230 J. Chem. Theory Comput., Vol. 2, No. 2, 2006 Parandekar and Tully 2. Two-Level Quantum Subsystem coefficients. Substituting eq 5 into the time-dependent In a previous publication,21 we derived a closed-form Schro¨dinger equation gives a set of coupled differential expression for the equilibrium mean energy of a two-level equations for the time-varying amplitudes cR(t), câ(t), and quantum subsystem coupled to an infinite number of classical cγ(t): particles via the Ehrenfest self-consistent field approximation. i We recast the amplitudes cR and c of quantum levels R and ˘ )- - 3‚ + 3‚ â cR pRcR q dRâcâ q dγRcγ (6a) â into two new variables i ) | |2 ) - | |2 c˘ )- c + q3‚d c - q3‚d c (6b) X câ 1 cR (1a) â p â â Râ R âγ γ and ˘ )- i - 3‚ + 3‚ cγ pγcγ q dγRcR q dâγcâ (6c) ) * + * Y cRcâ câcR (1b) where we have assumed, for simplicity of notation, that the The variables X and Y can be shown to behave as effective adiabatic wave functions Rq, âq, and γq are real-valued. The classical variables. 25 We derived a classical Liouville nonadiabatic coupling vector dRâ is given by equation for the probability distribution f(q,p,X,Y)ofthe ) 〈R |∇ 〉 positions q and momenta p of the classical particles and the dRâ q qâq (7) variables X and Y. We then obtained the steady-state solution of the Liouville equation, which produced the following Couplings between the other states are defined similarly. simple expression for the equilibrium mean energy of the Note that, because we have chosen the adiabatic representa- quantum subsystem in terms of the temperature, T,ofthe tion, there is no potential energy term coupling the quantum classical bath: levels. Since the Ehrenfest method is invariant to choice of representation,26 the results we derive here apply for any valid - â exp[ â/kBT] representation. From eq 6, we obtain the time derivatives of h)〈| |2〉 ) - E câ â kBT - - (2) 1 exp[ â/kBT] the populations. d where the energy of the lower quantum level R is taken to |c |2 )-q3‚d (c* c + c*c ) + q3‚d (c* c + c*c ) (8a) dt R Râ R â â R γR R γ γ R be zero, â is the energy of the upper level â, and kB is Boltzmann’s constant. It can be shown with some algebra d |c |2 ) q3‚d (c* c + c*c ) - q3‚d (c*c + c*c ) (8b) that the mean energy of the quantum subsystem produced dt â Râ R â â R âγ â γ γ â by Ehrenfest dynamics, as given by eq 2, is always greater than the desired Boltzmann energy d |c |2 ) q3‚d (c*c + c*c ) - q3‚d (c* c + c*c ) (8c) dt γ âγ â γ γ â γR R γ γ R exp[- /k T] h ) â â B EBOLTZ + - (3) Summing all parts of eq 8 demonstrates conservation of 1 exp[ â/kBT] norm. We define four independent effective classical phase 3. Three-Level Quantum Subsystem space variables for the quantum subsystem, U, S, W, and X, We first generalize this result to a three-level quantum in terms of the quantum amplitudes: subsystem and, later, to an N-level system. We follow the ) | |2 same procedure as above. The three-level quantum subsystem U câ (9) is described in an adiabatic basis. The wave functions Rq, S ) |c |2 (10) âq, and γq are the eigenfunctions of the quantum Hamiltonian γ for fixed classical positions q W ) c*c + c*c (11) R ) R â γ γ â Hq q R(q) q (4a) ) * + * ) X câcR cRcâ (12) Hqâq â(q) âq (4b) The variables U, S, W, and X are the independent variables γ ) (q) γ (4c) Hq q γ q required to characterize a three-state system. Whereas the The subscript q indicates that the quantum Hamiltonian and three complex-valued amplitudes introduce six variables, two | |2 its eigenfunctions depend parametrically on the classical are not independent. The quantity cR is determined by conservation of norm, and the variable positions q. The eigenenergies R(q), â(q), and γ(q), in general, are also functions of q; they are the adiabatic Y ) c*c + c* c (13) potential energy surfaces for states R, â, and γ, respectively. γ R R γ We express the wave function of the quantum subsystem at can be expressed as follows in terms of the independent R any time t as a linear combination of q, âq, and γq variables: ) R + + ψ(t) cR(t) q câ(t) âq cγ(t) γq (5) (x4US - W2x4U - 4U2 - 4US - X2 + WX Y ) (14) where cR(t), câ(t), and cγ(t) are the complex-valued expansion 2U Ehrenfest Mixed Quantum-Classical Dynamics J. Chem. Theory Comput., Vol. 2, No. 2, 2006 231 From eqs 8-12, we obtain equations for the time derivatives Nc - - 2 of the four effective classical variables U, S, W, and X: f(q,p,U,S,W,X) ) A e V(q)/kBT ∏ e pi /2mkBT g(U,S,W,X) (21) i)1 ˙)-3‚ + 3‚ U q dâγW q dRâX (15) where A is a normalization constant and ˙)3‚ - 3‚ S q dâγW q dγRY (16) 1 - g(U,S,W,X) ) (4US - W2) 1/2(4U - 4U2 - ˙) - 3‚ - 3‚ + 3‚ ( 2 W 2(U S)q dâγ q dγRX q dRâY π ( - ) U( - ) γ â x - 2 - 2 -1/2 - â R p 4US W (17) 4US X ) exp[ ] kBT S( - ) X˙)2(1 - 2U - S)q3‚d + q3‚d W - q3‚d Y ( γ R R Râ γR âγ exp[- ] exp(- ) kBT kBT (R - ) â x4U - 4U2 - 4US - X2 (18) p 1 - ) (4US - W2) 1/2(4U - 4U2 - 2 The two separate branches (() in eqs 17 and 18 can be π - 2 -1/2 - | |2 treated separately and, thereby, pose no mathematical dif- 4US X ) exp[ ( ∑ ci i)/kBT] (22) ficulty.
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